Structured backward error for palindromic polynomial eigenvalue problems. II: Approximate eigentriplets
Given an approximate eigentriplet for a palindromic polynomial matrix eigenvalue problem $P(\lambda)x = 0$, the paper tries to estimate the smallest perturbation of $P(\lambda)$ such that the given eigentriplet solves the perturbed problem exactly. It finds computable bounds for the structured backward errors where structured here means real symmetric or complex Hermitean matrices and similarly assumes only structured perturbations. This problem has applications in the vibration analysis of fast trains which invariably have both large and tiny magnitude eigenvalues. The authors' aim is to find the backward error of minimal norm that can be easily computed from the data. This optimization problem is first simplified via the QR decomposition of the given eigendata triple and then solved via detailed estimations of the thereby reduced number of variables. Numerical examples show the tightness of the found upper error bounds and two critical cases from earlier work on eigenpairs resurface here as well. The main complications for eigentriplets of palindromic matrix eigenproblems come from the doubling of coupled constraint equations in the optimization and bounding process when compared with the simpler and well understood eigenpair case. For Part I see [\textit{R.-C. Li} et al., Numer. Math. 116, No. 1, 95--122 (2010; Zbl 1213.65063)].
scientific article; zbMATH DE number 7036356
scientific article